Nonlinear Waves and Polarization in Diffusive Directed Particle Flow

Abstract

We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave solutions for weak diffusion by singular perturbation techniques and provide evidence for their existence also for stronger diffusion. We discuss different types of wave fronts and their composition to more complex patterns and illustrate their emergence from generic initial data by simulations. We also investigate the dependence of the wave velocities on the model parameters.

Global phase diagrams of run-and-tumble dynamics: equidistribution, waves, and blowup

“…To the right, regions of high density are analogously contained by barriers consisting of high u-concentrations. Traveling waves exist for larger values γ < 1/8, regardless of ε; see again [8,22], the earlier work [10] for such models applied to the dynamics of the cellular cytoskeleton, and [9].…”